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Abstract

Coherent diffraction imaging (CDI) is a method for reconstructing the complex-valued image of an object from diffraction intensities by using iterative phasing methods. X-ray ptychography is a scanning type of CDI using X-rays, allowing us to visualize the complex transmission function of an extended specimen. We here propose the use of the Kramers–Kronig relation (KKR) as an additional constraint in phase retrieval algorithms for multiple-energy X-ray ptychography using the absorption edge of a specific element. A numerical simulation showed that the speed of convergence was increased by using the improved algorithm with the KKR. We successfully demonstrated its usefulness in a proof-of-principle experiment at SPring-8. The present algorithm is particularly useful for imaging X-ray absorption fine structures of a specific element buried within thick samples by hard X-ray spectro-ptychography.

Figures (5)

(a) Reconstructed amplitude image without (left) and with (right) the KKR constraint. (b) Reconstructed phase image without (left) and with (right) the KKR constraint. The scale bar is 1 μm and X-ray energy is 6.557keV. (c) Iteration number dependence of the RMS error of the object image with and without using the KKR constraint. (d) Original (top) and reconstructed near-edge XAFS spectra without (middle) and with (bottom) using the KKR constraint. The spectra were extracted from the 40 × 40 nm2 region (2 × 2 pixels) indicated by the arrows in (a) and (b).

(a) Amplitude image reconstructed at 6.554 keV. The scale bar is 1 μm. (b) Near-edge XAFS spectrum of the sample measured in transmittance geometry using a focused X-ray beam (top). Near-edge XAFS spectra reconstructed by X-ray ptychography without (left) and with (right) using the KKR constraint. The spectra were extracted from the 40 × 40 nm2 regions (2 × 2 pixels) indicated by the arrows in (a). (c) Near-edge phase spectra at the same positions without (left) and with (right) using the KKR constraint.